Abstract: Alexei Vadimovich Pokrovskii was an outstanding mathematician, a scientist with very broad mathematical interests, and
a pioneer in the mathematical theory of systems with hysteresis. He died unexpectedly on September 1, 2010 at the age 62.
For the previous nine years he had been Professor and Head of Applied Mathematics at University College Cork in Ireland.

Abstract: Motivated by the fact that various experimental results yield strong confirmatory support for the hypothesis that the mixing of a wheat-flour dough is essentially a rate-independent process'', this paper examines how the mixing can be modelled using the rigorous mathematical framework developed to model an incremental time evolving deformation of an elasto-plastic material.
Initially, for the time evolution of a rate-independent elastic process, the concept is introduced of an "energetic solution'' [24] as the characterization for the rate-independent deformations occurring.
The framework in which it is defined is formulated in terms of a polyconvex stored energy density and a multiplicative decomposition of large deformations into elastic and nonelastic (plastic or viscous) components.
The mixing of a dough to peak dough development is then modelled as a sequence of incremental elasto-nonelastic deformations.
For such incremental processes, the existence of Sobolev solutions is guaranteed.
Finally, the limit passage to vanishing time increment leads to the existence of an energetic solution to our problem.

Abstract: In this paper we
consider Galerkin finite element discretizations of semilinear elliptic differential
inclusions that satisfy a relaxed one-sided Lipschitz condition.
The properties of the set-valued Nemytskii operators are discussed, and
it is shown that the solution sets of both, the continuous and the discrete
system, are nonempty, closed, bounded, and connected sets in $H^1$-norm.
Moreover, the solution sets of the Galerkin inclusion
converge with respect to the Hausdorff distance measured in $L^p$-spaces.

Abstract: One of the main paradigms of the theory of weakly interacting
chaotic systems is the absence of phase transitions in generic
situation. We propose a new type of multicomponent systems
demonstrating in the weak interaction limit both collective and
independent behavior of local components depending on fine
properties of the interaction. The model under consideration is
related to dynamical networks and sheds a new light to the problem
of synchronization under weak interactions.

Abstract: This paper is concerned with an optimal control problem for a system
of ordinary differential equations with rate independent hysteresis
modelled as a rate independent evolution variational inequality
with a closed convex constraint $Z\subset \mathbb{R}^m$.
We prove existence of optimal solutions as well as necessary optimality
conditions of first order. In particular, under certain regularity
assumptions we completely characterize the jump behaviour of the adjoint.

Abstract: If financial markets displayed the informational efficiency postulated in
the efficient markets hypothesis (EMH), arbitrage operations would be
self-extinguishing. The present paper considers arbitrage sequences in
foreign exchange (FX) markets, in which trading platforms and information
are fragmented. In [18,9] it was shown
that sequences of triangular arbitrage operations in FX markets containing
$4$ currencies and trader-arbitrageurs tend to display periodicity or grow
exponentially rather than being self-extinguishing. This paper extends the
analysis to $5$ or higher-order currency worlds. The key findings are that
in a $5$-currency world arbitrage sequences may also follow an exponential
law as well as display periodicity, but that in higher-order currency
worlds a double exponential law may additionally apply. There is an
``inheritance of instability'' in the higher-order currency worlds.
Profitable arbitrage operations are thus endemic rather that displaying
the self-extinguishing properties implied by the EMH.

Abstract: The ``new consensus'' DSGE(dynamic stochastic general equilibrium) macroeconomic model has microfoundations provided by a single representative agent. In this model shocks to the economic environment do not have any lasting effects. In reality adjustments at the micro level are made by heterogeneous agents, and the aggregation problem cannot be assumed away. In this paper we show that the discontinuous adjustments made by heterogeneous agents at the micro level mean that shocks have lasting effects, aggregate variables containing a selective, erasable memory of the shocks experienced. This hysteresis framework provides foundations for the post-Walrasian analysis of macroeconomic systems.

Abstract: Following the approach of [22], we derive a system of Fokker-Planck
equations to model a stock-market in which hysteretic agents can take
long and short positions. We show numerically that the resulting
mesoscopic model has rich behaviour, being hysteretic at the mesoscale
and displaying bubbles and volatility clustering in particular.

Abstract: Identification of biological models is often complicated by the fact
that the available experimental data from field measurements is
noisy or incomplete. Moreover, models may be complex, and contain a
large number of correlated parameters. As a result, the parameters
are poorly identified by the data, and the reliability of the model
predictions is questionable. We consider a general scheme for
reduction and identification of dynamic models using two modern
approaches, Markov chain Monte Carlo sampling methods together with
asymptotic model reduction techniques. The ideas are illustrated
using a simple example related to bio-medical applications: a model
of a generic receptor. In this paper we want to point out what the
researchers working in biological, medical, etc., fields should look
for in order to identify such problematic situations in modelling,
and how to overcome these problems.

Abstract: This paper proposes an approach to investigate bifurcation of periodic solutions to functional-differential equations of neutral type with a small delay and a small periodic perturbation from the limit cycle under the assumption that there exists adjoint Floquet solutions to the linearized equation.

Abstract: Discrete-time discrete-state random Markov chains with a tridiagonal
generator are shown to have a random attractor consisting of singleton
subsets, essentially a random path, in the simplex of probability vectors.
The proof uses the Hilbert projection metric and the fact that the linear
cocycle generated by the Markov chain is a uniformly contractive mapping
of the positive cone into itself. The proof does not involve probabilistic
properties of the sample path $\omega$ and is thus equally valid in the
nonautonomous deterministic context of Markov chains with, say,
periodically varying transitions probabilities, in which case the
attractor is a periodic path.

Abstract: We consider a scalar fast differential equation which is periodically driven by a slowly varying input.
Assuming that the equation depends on $n$ scalar parameters, we present simple sufficient conditions
for the existence of a periodic canard solution, which, within a period, makes $n$ fast transitions
between the stable branch and the unstable branch of the folded critical curve.
The closed trace of the canard solution on the plane of the slow input variable and the fast phase variable has $n$
portions elongated along the unstable branch of the critical curve. We show that the length of these portions and the
length of the time intervals of the slow motion separated by the short time intervals of fast transitions
between the branches are controlled by the parameters.

Abstract: Suggested by conversations in 1991 (Mark Krasnosel'skiĭ and Aleksei Pokrovskiĭ with TIS), this paper generalizes earlier work [7] of theirs
by defining a setting of hybrid systems with isotone switching rules
for a partially ordered set of modes and then obtaining a periodicity
result in that context. An application is given to a partial
differential equation modeling calcium release and diffusion in
cardiac cells.

Abstract: The paper deals with the study of the relation between the Andronov--Hopf bifurcation,
the canard explosion and the critical phenomena for the van der Pol's type system of singularly
perturbed differential equations. Sufficient conditions for the limit cycle birth bifurcation
in the case of the singularly perturbed systems are investigated.
We use the method of integral manifolds and canards techniques
to obtain the conditions under which the system possesses
the canard cycle.
Through the application to some chemical and optical models
it is shown that the canard point should be considered as the critical value
of the control parameter.

Abstract: The existence of canard cascades is studied in the paper as a
problem of gluing of stable and unstable one-dimensional slow
invariant manifolds at turning points. This way of looking is made
feasible to establish the existence of canard cascades, that can
be considered as a generalization of canards. A further
development of this approach, with applications to the van der Pol
equation and a problem of population dynamics, is contained in the
paper.

Abstract: We establish a large deviation principle for stochastic
differential equations with averaging in the case when all
coefficients of the fast component depend on the slow one,
including diffusion.

Abstract: Electromagnetic processes in a ferromagnetic conductor
(e.g., an electric transformer) are here described by
coupling the Maxwell equations with nonlinear constitutive laws of the form
$$
\vec B \in \mu_0\vec H + {\mathcal M}(x) \vec H/|\vec H|,
\qquad
\vec J = \sigma(x) \big( \vec E + \vec E_a(x,t) + h(x)\vec J \!\times\! \vec B \big).
$$
Here $\vec E_a$ stands for an applied electromotive force; the saturation
${\mathcal M}(x)$, the conductivity $\sigma(x)$ and the Hall coefficient $h(x)$ are also prescribed.
The first relation accounts for hysteresis-free ferromagnetism,
the second one for the Ohm law and the Hall effect.
This model leads to the formulation of an initial-value problem for a doubly-nonlinear parabolic-hyperbolic system in the whole $R^3$. Existence of a weak solution is proved, via approximation by time-discretization, derivation of a priori estimates, and passage to the limit.
This final step rests upon a time-dependent extension of the Murat and Tartar div-curl lemma, and on compactness by strict convexity.

Abstract: We prove that the sweeping process on a "regular" class of
convex sets is equicontinuous. Classes of polyhedral sets with
a given finite set of normal vectors are regular, as well as
classes of uniformly strictly convex sets. Regularity is
invariant to certain operations on classes of convex sets such
as intersection, finite union, arithmetic sum and affine
transformation.

Abstract: We consider an "elastic'' version of the statistical mechanical monomer-dimer problem on the $n$-dimensional integer lattice. Our setting includes the classical "rigid'' formulation as a special case and extends it by allowing each dimer to consist of particles at arbitrarily distant sites of the lattice, with the energy of interaction between the particles in a dimer depending on their relative position. We reduce the free energy of the elastic dimer-monomer (EDM) system per lattice site in the thermodynamic limit to the moment Lyapunov exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value and covariance function are the Boltzmann factors associated with the monomer energy and dimer potential. In particular, the classical monomer-dimer problem becomes related to the MLE of a moving average GRF. We outline an approach to recursive computation of the partition function for "Manhattan'' EDM systems where the dimer potential is a weighted $l_1$-distance and the auxiliary GRF is a Markov random field of Pickard type which behaves in space like autoregressive processes do in time. For one-dimensional Manhattan EDM systems, we compute the MLE of the resulting Gaussian Markov chain as the largest eigenvalue of a compact transfer operator on a Hilbert space which is related to the annihilation and creation operators of the quantum harmonic oscillator and also recast it as the eigenvalue problem for a pantograph functional-differential equation.